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arXiv · 2310.02551

Apex Graphs and Cographs

Jagdeep Singh·Vaidy Sivaraman·Thomas Zaslavsky
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Abstract

A class $\mathcal{G}$ of graphs is called hereditary if it is closed under taking induced subgraphs. We denote by $\mathcal{G}^\mathrm{apex}$ the class of graphs $G$ that contain a vertex $v$ such that $G-v$ is in $\mathcal{G}$. We prove that if a hereditary class $\mathcal{G}$ has finitely many forbidden induced subgraphs, then so does $\mathcal{G}^\mathrm{apex}$. The hereditary class of cographs consists of all graphs $G$ that can be generated from $K_1$ using complementation and disjoint union. A graph is an apex cograph if it contains a vertex whose deletion results in a cograph. Cographs are precisely the graphs that do not have the $4$-vertex path as an induced subgraph. Our main result finds all such forbidden induced subgraphs for the class of apex cographs.

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Jagdeep Singh, Vaidy Sivaraman, Thomas Zaslavsky. 2023-10-04. Apex Graphs and Cographs. https://arxiv.org/abs/2310.02551

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